Nontrivial Recurrence Research Articles

There Exist Transitive Piecewise Smooth Vector Fields on $$\mathbb {S}^2$$ but Not Robustly Transitive

It is well known that smooth (or continuous) vector fields cannot be topologically transitive on the sphere $\S^2$. Piecewise-smooth vector fields, on the other hand, may present non-trivial recurrence even on $\S^2$. Accordingly, in this paper the existence of topologically transitive piecewise-smooth vector fields on $\S^2$ is proved, see Theorem \ref{teorema-principal}. We also prove that transitivity occurs alongside the presence of some particular portions of the phase portrait known as {\it sliding region} and {\it escaping region}. More precisely, Theorem \ref{main:transitivity} states that, under the presence of transitivity, trajectories must interchange between sliding and escaping regions through tangency points. In addition, we prove that every transitive piecewise-smooth vector field is neither robustly transitive nor structural stable on $\S^2$, see Theorem \ref{main:no-transitive}. We finish the paper proving Theorem \ref{main:general} addressing non-robustness on general compact two-dimensional manifolds.

Variation in recurrence rate and overall survival (OS) outcomes by disease stage and incremental impact of time to recurrence on OS in localized renal cell carcinoma (RCC).

4543 Background: Prior research shows that post-nephrectomy recurrence is correlated with significantly increased mortality in patients (pts) with intermediate-high (int-high) risk and high risk RCC. However, no study has quantified the incremental OS associated with increased time to recurrence (ToR). Additionally, limited evidence is presented on the variation of recurrence rate and OS by stage of RCC. Methods: The SEER-Medicare database (2007–2016) was used in this retrospective observational study. Post-nephrectomy pts with newly diagnosed, int-high risk (pT2 N0 high grade, pT3 N0 any grade) or high-risk (pT4 N0 any grade, pT any N1 any grade) RCC were identified and stratified based on tumor stage and grade. Grade was defined based on Furhman grading system and was reported in SEER Registry. Post-nephrectomy recurrence free rates and OS by disease stage were described using Kaplan-Meier analyses. OS from nephrectomy in pts with recurrence vs pts without recurrence was compared by disease stage. Multivariable regression analysis was used to quantify the incremental OS associated with increased ToR post-nephrectomy in all pts with recurrence. Results: 643 pts met the inclusion criteria (269 with vs 374 without recurrence; median follow-up: 23 months). The mean age was 75.5 years (yrs), 61% male, and 86% white. Results presented in the table showed wide variance in 5-yr recurrence-free rate in int-high risk group (28%-63%) and indicated substantial risk of disease recurrence in all subgroups of int-high risk pts. Among those with T3 Grade 1-2, T3 Grade 3, and T3 Grade 4 disease, pts with recurrence had significantly higher risk of death than those without (all ps<0.05;). Results for pts with T2 and T4 disease were not presented due to small sample size. Multivariable regression analysis indicated that 1 additional yr of ToR was associated with 0.73 additional yrs (8.8 additional months) of OS post nephrectomy (95% CI: 0.40, 1.05 yrs; p<0.001). Conclusions: The non-trivial recurrence rates observed in pts with T3 Grade 1-2 and T3 Grade 3 RCC highlighted the substantial risk of recurrence and unmet needs in the int-high risk RCC patients. This study also confirms the incremental nature of the association of ToR and OS in patients with int-high and high risk localized RCC. These findings highlight the need for effective early intervention with adjuvant treatments in int-high and high risk post-nephrectomy RCC pts. [Table: see text]

Ramanujan-type systems of nonlinear ODEs for [formula omitted] and [formula omitted

This paper aims to introduce two systems of nonlinear ordinary differential equations whose solution components generate the graded algebra of quasi-modular forms on Hecke congruence subgroups Γ0(2) and Γ0(3). Using these systems, we provide the generated graded algebras with an sl2(ℂ)-module structure. As applications, we introduce Ramanujan-type tau functions for Γ0(2) and Γ0(3), and obtain some interesting and non-trivial recurrence and congruence relations.

Limit Cycles of Piecewise Smooth Differential Equations on Two Dimensional Torus

In this paper we study the limit cycles of some classes of piecewise smooth vector fields defined in the two dimensional torus. The piecewise smooth vector fields that we consider are composed by linear, Ricatti with constant coefficients and perturbations of these one, which are given in (3). Considering these piecewise smooth vector fields we characterize the global dynamics, studying the upper bound of number of limit cycles, the existence of non-trivial recurrence and a continuum of periodic orbits. We also present a family of piecewise smooth vector fields that posses a finite number of fold points and, for this family we prove that for any 2k number of limit cycles there exists a piecewise smooth vector fields in this family that presents k number of limit cycles and prove that some classes of piecewise smooth vector fields presents a non-trivial recurrence or a continuum of periodic orbits.

On the Closing Lemma for planar piecewise smooth vector fields

A large number of papers deal with “Closing Lemmas” for Cr-vector fields (and Cr-diffeomorphisms). Here, we introduce this subject and formalize the terminology about nontrivially recurrent points and nonwandering points for the context of planar piecewise smooth vector fields. A global bifurcation analysis of a special family of piecewise smooth vector fields presenting a nonwandering set with nontrivial recurrence is performed. As a consequence, we are able to say that the Classical and the Improved Closing Lemmas are false for this scenario.

A structure theorem for foliations on non-compact 2-manifolds

In this paper we study the structure of non-trivial recurrence leaves. The obtained results have been shown to be very useful in the smoothing of the continuous singular foliations on arbitrary genus two-manifolds [Lo1]. From the famous example of Denjoy [De], it is clear that it is very important to understand the dynamical structure of non-trivial recurrence. For compact two-manifolds, the study of non-trivial recurrence together with the smoothing problems for orientable singular foliations was carried out by C. Gutierrez [Gu1]. On the other hand, recent work shows that infinite genus surfaces can support orientable foliations whose recurrent leaves have dynamics which can not appear on compact surfaces (see [Gu-He-Lo]). Besides, the existence of minimal foliations on those surfaces was already proven by J. Beniere (see [Be]). Our main theorem generalize, to orientable singular foliations on two-manifolds of infinite genus, C. Gutierrez’s structure theorem [Gu1]. To state the theorem, we need some definitions. Let T : R/Z→ R/Z be a map whose domain of definition (Dom(T )) and image (Im(T )) are open and dense subsets of R/Z. We say that T is a generalized interval exchange transformation(or shortly a GIET) if T takes homeomorphically each connected component of its domain of definition onto a connected component of its image. A GIET is said to be affine (resp. isometric) if it restricted to every connected component of its domain of definition, is affine (resp. isometric). If T is an isometric GIET such that R/Z Dom(T ) is at most finite, then we shall say that T is a standard interval exchange transformation (standard IET).

A study of a generalization of a card problem

In this paper, we study a card problem, and its generalization, related to a computer based Cribbage player, namely, how many ways are there to select 12 Poker cards out of a deck of 52 such that their total face value exceeds 31? As an immediate application, the answer to the above question provides useful information for designing and enhancing computer based game playing software for Cribbage and other similar card games. Moreover, the analysis of the general problem also leads to a solution to a problem of “sampling without replacement”. Besides presenting a combinatorial solution to the general problem, and discussing its “expressive complexity” via its relationship with the well-known integer partition problem, we also provide a computer solution, implemented in Prolog, based on a non-trivial recurrence relation. Relying on the latter result, we then carry out preliminary analysis of the computational aspects of some of the rules governing the Cribbage game.

From local to global behavior in competitive Lotka-Volterra systems

In this paper we exploit the linear, quadratic, monotone and geometric structures of competitive Lotka-Volterra systems of arbitrary dimension to give geometric, algebraic and computational hypotheses for ruling out non-trivial recurrence. We thus deduce the global dynamics of a system from its local dynamics. The geometric hypotheses rely on the introduction of a split Liapunov function. We show that if a system has a fixed point p ∈ int ⁡ R + n p\in \operatorname {int}{{\mathbf R}^n_+} and the carrying simplex of the system lies to one side of its tangent hyperplane at p p , then there is no nontrivial recurrence, and the global dynamics are known. We translate the geometric hypotheses into algebraic hypotheses in terms of the definiteness of a certain quadratic function on the tangent hyperplane. Finally, we derive a computational algorithm for checking the algebraic hypotheses, and we compare this algorithm with the classical Volterra-Liapunov stability theorem for Lotka-Volterra systems.

Linear dynamically varying LQ control of nonlinear systems over compact sets

Linear-quadratic controllers for tracking natural and composite trajectories of nonlinear dynamical systems evoluting over compact sets are developed. Typically, such systems exhibit complicated dynamics, i.e., have nontrivial recurrence. The controllers, which use small perturbations of the nominal dynamics as input actuators, are based on modeling the tracking error as a linear dynamically varying (LDV) system. Necessary and sufficient conditions for the existence of such a controller are linked to the existence of a bounded solution to a functional algebraic Riccati equation (FARE). It is shown that, despite the lack of continuity of the asymptotic trajectory relative to initial conditions, the cost to stabilize about the trajectory, as given by the solution to the FARE, is continuous. An ergodic theory method for solving the FARE is presented. Furthermore, it is shown that wrapping the LDV controller around the nonlinear system secures a stable tracking dynamics. Finally, an example of controlling the Henon map to follow an aperiodic orbit is presented.

On Cherry flows

AbstractThe purpose of this research is to describe all smooth vector fields on the torus T2 with a finite number of singularities, no periodic orbits and no saddleconnections. In this paper we are able to complete the description within the class of vector fields which are area contracting near all singularities. In particular we give a large class of analytic vector fields on the torus T2 which have non-trivial recurrence and also sinks.

Differentiability and topology of labyrinths in the disc and annulus

IT IS WELL-KNOWN that vector fields on the disc, or more generally on a planar surface, do not exhibit complicated global phenomena: by the Poincare-Bendixson theorem, every recurrent orbit is either a singularity or a simple closed curve. The situation is completely different for line fields. In particular, it was shown in [6] that it is possible for an orbit of a line field to lose itself in a labyrinth, i.e. to enter a disc and stay inside for ever without either going to a singularity or spiralling towards a compact cycle. Line fields in the disc and the annulus were studied and classified in [6], under the assumption that the singularities are thorns and tripods. It was also asked in [6] to what extent line fields with nontrivial recurrence can be differentiable. We intend in this paper first to partially answer this question, then to extend the classification in [6] to the case when singularities are thorns and saddles with any number of prongs. First we discuss differentiability. There are several natural definitions of C’differentiability for singular line fields (even for orientable ones). First one can require the line field to be C’ in the complement of the singularities; it was pointed out in [6] that being C” in this sense does not restrict the dynamics of the foliation. Another notion of differentiability imposes strong C’ local models near the singularities (see 92). Assuming C2-differentiability in this sense, we prove that the Poincart-Bendixson theorem continues to hold; that is, every orbit entering a disc and staying inside has to go to a singularity or spiral towards a compact cycle. The proof is inspired in part by Denjoy’s theorem about C2 vector fields on the torus. This is in sharp contrast with the fact due to Cherry [l], that on the torus minus a disc there are vector fields (Cm in the above sense) where every orbit coming in from the boundary stays inside forever and does not spiral towards a compact cycle. A very interesting question is whether the Poincare-Bendixson theorem continues to hold for line fields with hyperbolic-type singularities; by this we mean that the natural holonomy maps obtained by following a separatrix through a singularity are of the same form as for hyperbolic vector fields. In $3 and $4 of this paper, we extend the results of [6] to line fields with thorns and n-prong saddles. For the disc, the situation is not essentially different from the case of thorns and tripods. However, for the annulus, the study is quite different and more complicated. In both cases, we obtain a topological structure theorem for the foliation. In several instances the proofs we give simplify arguments in [6].

Smoothing continuous flows on two-manifolds and recurrences

AbstractLet φ: ℝ × M → M be a continuous flow on a compact C∞ two-manifold M. It is proved that there exists a C1 flow ψ on M which is topologically equivalent to φ, and that the following conditions are equivalent:(a) any minimal set of φ is trivial;(b) φ is topologically equivalent to a C2 flow;(c) φ is topologically equivalent to a C∞ flow.Also proved is a structure and an existence theorem for continuous flows with non-trivial recurrence.

The structure of flows exhibiting nontrivial recurrence on two-dimensional manifolds

The aim of this paper is to describe the orbit structure of flows exhibiting nontrivial recurrence on closed, two-dimensional manifolds, insofar as that structure is determined by the nontrivially recurrent orbits. The description involves two steps: the first is to decompose the manifold into certain pieces, each of which contains at most one recurrent orbit closure; and the second is to describe the flow on those pieces containing a recurrent orbit closure in terms of a transitive flow on the same submanifold (compactidied). The two steps are presented in the reverse order-Theorem 1 formalises the second step and Theorem 2, which follows easily from Theorem 1, the first. Theorem 2 is similar to Theorem B of Gutierrez [8]; one advantage of the present approach is that Theorem 1 enables one to apply the work of Aranson and Grines [ 1,2] to the problem of classifying flows on twomanifolds up to topological equivalence. This classification is outlined briefly in the final section. For reasons which will become clear, attention is in general restricted to flows with finitely many singular points.

A recurrence relation for the polarization in elastic p-p scattering in the framework of a parton model

Recent studies have shown that it is possible to account for the general features of large angle elastic proton-proton differential cross sections in a model where each constituent of one proton scatters from at least one constituent of the other proton in such a way that all constituents remain near their mass shell. Each of the parton-parton scatters is assumed to occur at approximately the same angle, so that the outgoing partons can easily recombine to form the two final state protons. In the present article we show that such a model, with additional assumptions regarding the scattering mechanisms at small | t|, leads to a non-trivial recurrence relation which is consistent with the structure recently reported in the high energy p-p polarization and fixed angle differential cross section.